Computational Optimization and Applications

, Volume 62, Issue 3, pp 717–759 | Cite as

Convex reformulations for solving a nonlinear network design problem

  • Jesco Humpola
  • Armin FügenschuhEmail author


We consider a nonlinear nonconvex network design problem that arises, for example, in natural gas or water transmission networks. Given is such a network with active and passive components, that is, valves, compressors, control valves (active) and pipelines (passive), and a desired amount of flow at certain specified entry and exit nodes in the network. The active elements are associated with costs when used. Besides flow conservation constraints in the nodes, the flow must fulfill nonlinear nonconvex pressure loss constraints on the arcs subject to potential values (i.e., pressure levels) in both end nodes of each arc. The problem is to compute a cost minimal setting of the active components and numerical values for the flow and node potentials. We examine different (convex) relaxations for a subproblem of the design problem and benefit from them within a branch-and-bound approach. We compare different approaches based on nonlinear optimization numerically on a set of test instances.


Nonlinear network flow Mixed-integer nonlinear programming Relaxations Network design 



We are grateful to Open Grid Europe GmbH (OGE, Essen/Germany) for supporting our work. The second coauthor conducted parts of this research under a Konrad-Zuse-Fellowship. We thank two anonymous referees for their various helpful comments on our manuscript.


  1. 1.
    Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Babonneau, F., Nesterov, Y., Vial, J.-P.: Design and operations of gas transmission networks. Oper. Res. 60(1), 34–47 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Collins, M., Cooper, L., Helgason, R., Kennington, J., LeBlanc, L.: Solving the pipe network analysis problem using optimization techniques. Manag. Sci. 24(7), 747–760 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. SIAM, Philadelphia (2000)zbMATHCrossRefGoogle Scholar
  5. 5.
    CPLEX: User’s Manual for CPLEX. IBM Corporation, 12.1 edition, Armonk, USA (2011)Google Scholar
  6. 6.
    De Wolf, D.: Mathematical properties of formulations of the gas transmission problem. Submitted to RAIRO Oper. Res. (2004).
  7. 7.
    De Wolf, D., Bakhouya, B.: The gas transmission problem when the merchant and the transport functions are disconnected. Technical Report 01/01, Ieseg, Université catholique de Lille, HEC Ecole de Gestion de l’ULG (2007)Google Scholar
  8. 8.
    De Wolf, D., Bakhouya, B.: Optimal dimensioning of pipe networks: the new situation when the distribution and the transportation functions are disconnected. Technical Report 07/02, Ieseg, Université catholique de Lille, HEC Ecole de Gestion de l’ULG (2008)Google Scholar
  9. 9.
    De Wolf, D., Bakhouya, B.: Solving gas transmission problems by taking compressors into account., September 2008. Submitted to 4OR
  10. 10.
    De Wolf, D., Smeers, Y.: Optimal dimensioning of pipe networks with application to gas transmission networks. Oper. Res. 44(4), 596–608 (1996)zbMATHCrossRefGoogle Scholar
  11. 11.
    De Wolf, D., Smeers, Y.: The gas transmission problem solved by an extension of the simplex algorithm. Manag. Sci. 46(11), 1454–1465 (2000)zbMATHCrossRefGoogle Scholar
  12. 12.
    Dembo, R.S., Mulvey, J.M., Zenios, S.A.: Large-scale nonlinear network models and their application. Oper. Res. 37(3), 353–372 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Fügenschuh, A., Homfeld, H., Schülldorf, H., Vigerske, S.: Mixed-integer nonlinear problems in transportation applications. In: Rodrigues, H., et al. (eds.) Proceedings of the 2nd International Conference on Engineering Optimization (CD-ROM) (2010)Google Scholar
  14. 14.
    Geißler, B., Martin, A., Morsi, A.: LaMaTTO++. Information available at, February 2015
  15. 15.
    Humpola, J., Fügenschuh, A., Koch, T.: A New Class of Valid Inequalities for Nonlinear Network Design Problems. OR Spectrum, online available (2015)Google Scholar
  16. 16.
    Humpola, J., Fügenschuh, A., Lehmann, T.: A primal heuristic for optimizing the topology of gas networks based on dual information. EURO J. Comput. Optim. 3(1), 53–78 (2015)zbMATHCrossRefGoogle Scholar
  17. 17.
    Karush, W.: Minima of functions of several variables with inequalities as side constraints. Master’s thesis (1939)Google Scholar
  18. 18.
    Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms. Springer, Berlin (2007)Google Scholar
  19. 19.
    Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the 2nd Berkley Symposium on Mathematical Statistics and Probability, pp. 481–493. University Press, Berkley, California (1951)Google Scholar
  20. 20.
    Maugis, J.J.: Etude de réseaux de transport et de distribution de fluide. RAIRO Oper. Res. 11(2), 243–248 (1977)Google Scholar
  21. 21.
    Nemhauser, G.L., Wolsey, L.A.: Integer programming, Chap. 6. In: Nemhauser, G.L., Rinnooy Kan, A.H.G., Todd, M.J. (eds.) Optimization, pp. 447–527. Elsevier, Amsterdam (1989)CrossRefGoogle Scholar
  22. 22.
    Oldham, J.: Combinatorial approximation algorithms for generalized flow problems. In: Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms SODA’99, pp. 704–714 (1999)Google Scholar
  23. 23.
    Pfetsch, M., Fügenschuh, A., Geißler, B., Geißler, N., Gollmer, R., Hiller, B., Humpola, J., Koch, T., Lehmann, T., Martin, A., Morsi, A., Rövekamp, J., Schewe, L., Schmidt, M., Schultz, R., Schwarz, R., Schweiger, J., Stangl, C., Steinbach, M., Vigerske, S., Willert, B.: Validation of nominations in gas network optimization: models, methods, and solutions. Optim. Methods Softw. 30(1), 15–53 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Raghunathan, A.U.: Global optimization of nonlinear network design. SIAM J. Optim. 23(1), 268–295 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Sherali, H.D., Smith, E.P.: An optimal replacement-design model for a reliable water distribution network system. In: Coulbeck, Bryan (ed.) Integrated Computer Applications in Water Supply, vol. 1, pp. 61–75. Wiley, New York (1994)Google Scholar
  26. 26.
    Smith, E.M.B., Pantelides, C.C.: A symbolic reformulation/spatial branch-and-bound algorithm for the global optimization of nonconvex MINLPs. Comput. Chem. Eng. 23, 457–478 (1999)CrossRefGoogle Scholar
  27. 27.
    Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99(3), 563–591 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Vigerske, S.: Decomposition in Multistage Stochastic Programming and a Constraint Integer Programming Approach to Mixed-Integer Nonlinear Programming. PhD thesis, Humboldt-Universität zu Berlin (2012)Google Scholar
  30. 30.
    Wächter, A., Biegler, L.T.: On the implementation of a primal–dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Zuse Institute BerlinBerlinGermany
  2. 2.Helmut Schmidt University/University of the Federal Armed Forces HamburgHamburgGermany

Personalised recommendations